Exponential Functions: Reading Word Problems

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What is an exponential function?

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An exponential function is a mathematical function of the form f(x) = a * b^x, where 'a' is a constant, 'b' is a positive real number, and 'x' is the exponent. It describes growth or decay processes.

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What does it mean for a function to exhibit exponential growth?

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Exponential growth occurs when the growth rate of a value is proportional to its current value, leading to rapid increases over time. For example, if a population doubles every year, it is experiencing exponential growth.

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What does it mean for a function to exhibit exponential decay?

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Exponential decay refers to a decrease in a quantity at a rate proportional to its current value, leading to a rapid decline. An example is radioactive decay, where the amount of substance decreases over time.

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How do you identify an exponential function from a word problem?

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Look for keywords indicating growth or decay, such as 'doubles', 'triples', 'halves', or 'decreases by a percentage'. Also, check for phrases that suggest a constant rate of change.

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What is the general form of an exponential growth equation?

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The general form is y = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

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What is the general form of an exponential decay equation?

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The general form is y = a(1 - r)^t, where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

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How do you solve a word problem involving exponential growth?

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1. Identify the initial amount, growth rate, and time period. 2. Use the exponential growth formula. 3. Substitute the values and solve for the unknown.

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How do you solve a word problem involving exponential decay?

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1. Identify the initial amount, decay rate, and time period. 2. Use the exponential decay formula. 3. Substitute the values and solve for the unknown.

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What is the significance of the base 'b' in an exponential function?

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The base 'b' determines the rate of growth or decay. If b > 1, the function represents growth; if 0 < b < 1, it represents decay.

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What is a real-world example of exponential growth?

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A common example is compound interest in finance, where the amount of money grows exponentially over time based on the interest rate.